Question: Solve for $x$ and $y$ using elimination. $\begin{align*}2x+4y &= 7 \\ 4x+3y &= 8\end{align*}$
We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $1$ $\begin{align*}-4x-8y &= -14\\ 4x+3y &= 8\end{align*}$ Add the top and bottom equations. $-5y = -6$ Divide both sides by $-5$ and reduce as necessary. $y = \dfrac{6}{5}$ Substitute $\dfrac{6}{5}$ for $y$ in the top equation. $2x+4( \dfrac{6}{5}) = 7$ $2x+\dfrac{24}{5} = 7$ $2x = \dfrac{11}{5}$ $x = \dfrac{11}{10}$ The solution is $\enspace x = \dfrac{11}{10}, \enspace y = \dfrac{6}{5}$.